This paper investigates the gradient flow structure, well-posedness, and asymptotic behavior of the Fokker-Planck equation defined on locally uniformly finite graphs, which is highly non-trivial compared with the finite case. We first construct a 2-Wasserstein-type metric and gradient flow equation in the probability density space associated with the underlying graphs. Then, we prove the global existence of solution to the Fokker-Planck equation using a novel approach that differs significantly from the methods applied in the finite case. We also demonstrate that the solution converges to the Gibbs distribution in the $\ell^{r}(V,\bmπ)$ norm with $r\in [2,\infty]$, by using the indicator set partitioning method. To the best of our knowledge, this work seems the first result on the study of Wasserstein-type metrics and the Fokker-Planck equation in probability density spaces defined on infinite graphs.